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Lesson 7-2 Solving Systems by SUBSTITUTION, page 347

Lesson 7-2 Solving Systems by SUBSTITUTION, page 347. Objective: To solve systems of equations using substitution. REVIEW. Name one way we’ve learned to solve a system of equations. What are the three possible solutions to solving a system by graphing?

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Lesson 7-2 Solving Systems by SUBSTITUTION, page 347

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  1. Lesson 7-2 Solving Systems by SUBSTITUTION, page 347 Objective: To solve systems of equations using substitution.

  2. REVIEW • Name one way we’ve learned to solve a system of equations. • What are the three possible solutions to solving a system by graphing? • Lines intersect in one point. This point will “work” in BOTH equations when substituted. • Lines are parallel (no solution) • Lines are collinear (infinitely many solutions)

  3. Think about this… • Do you think graphing is the only way to solve a system? • What if the equations are not “easy” to graph? • We can also use the SUBSTITUION Method to solve systems.

  4. Steps for using SUBSTITUTION • Solve one equation for one variable. (Hint: Look for an equation already solved for one of the variables or for a variable with a coefficient of 1.) • Substitute into the other equation. • Solve this equation to find a value for a variable. • Substitute again to find the value of the other variable.

  5. Solving Systems Using Substitution Solve using substitution. y= 2x + 2 y= –3x + 4 Step 1:Write an equation containing only one variable and solve. y= 2x + 2Start with one equation. –3x + 4 = 2x + 2Substitute –3x + 4 for y in that equation. 4 = 5x + 2 Add 3x to each side. 2 = 5x Subtract 2 from each side. 0.4 = x Divide each side by 5. Step 2:Solve for the other variable. y = 2(0.4) + 2 Substitute 0.4 for x in either equation. y = 0.8 + 2Simplify. y = 2.8 7-2

  6. Check: See if (0.4, 2.8) satisfies y= –3x + 4 since y= 2x + 2 was used in Step 2. 2.8 –3(0.4) + 4 Substitute(0.4, 2.8) for (x, y) in the equation. 2.8 –1.2 + 4 2.8 = 2.8 Solving Systems Using Substitution (continued) Since x= 0.4 and y = 2.8, the solution is (0.4, 2.8). 7-2

  7. Solving Systems Using Substitution Solve using substitution. –2x + y = –1 4x + 2y = 12 Step 1: Solve the first equation for y because it has a coefficient of 1. –2x + y = –1 y = 2x –1Add 2x to each side. Step 2: Write an equation containing only one variable and solve. 4x + 2y = 12 Start with the other equation. 4x + 2(2x –1) = 12 Substitute 2x –1 for y in that equation. 4x + 4x –2 = 12 Use the Distributive Property. 8x= 14 Combine like terms and add 2 to each side. x= 1.75 Divide each side by 8. 7-2

  8. Solving Systems Using Substitution (continued) Step 3: Solve for y in the other equation. –2(1.75)+ y = 1 Substitute 1.75 for x. –3.5+ y = –1 Simplify. y = 2.5 Add 3.5 to each side. Since x= 1.75 and y = 2.5, the solution is (1.75, 2.5). 7-2

  9. Ex. 1) Solve using substitution. y = 2x y = x + 7 Ex. 2) Solve using substitution. 6y + 8x = 28 3 = 2x – y EXAMPLES 1 & 2, page 348

  10. You try. • Solve the system using substitution. y = 4x - 8 y = 2x + 10

  11. POSSIBLE SOLUTIONS • Answer is a point, (x, y). • If variables cancel out and you get a true statement, the solution is all real numbers. • If variables cancel out and you get a false statement, there is no solution.

  12. Ex.) y = 2x 2x + 5y = -12 Solution: (-1,-2) Ex.) 2y = -3x 4x + y = 5 Solution: (2, -3) More EXAMPLES FOR YOU!Use SUBSTITUTION to solve.

  13. Ex.) x + y = 6 3x + y = 15 Solution: (9/2, 3/2) Need one more?

  14. Ex.) x + y = 5 2x +2y = 10 Solution: Infinitely many Need one more?

  15. Ex.) y = x + 2 3x – 3y = 10 Solution: no solution Need one more?

  16. Let v= number of vans and c = number of cars. Drivers v + c = 5 Persons 7 v + 5 c = 31 Solving Systems Using Substitution A youth group with 26 members is going to the beach. There will also be five chaperones that will each drive a van or a car. Each van seats 7 persons, including the driver. Each car seats 5 persons, including the driver. How many vans and cars will be needed? Solve using substitution. Step 1: Write an equation containing only one variable. v+ c = 5 Solve the first equation for c. c = –v + 5 7-2

  17. Solving Systems Using Substitution (continued) Step 2: Write and solve an equation containing the variable v. 7v+ 5c = 31 7v+ 5(–v + 5) = 31 Substitute –v + 5 for c in the second equation. 7v– 5v + 25 = 31 Solve for v. 2v + 25 = 31 2v = 6 v = 3 Step 3: Solve for c in either equation. 3 + c = 5 Substitute 3 for v in the first equation. c = 2 7-2

  18. Real-World ConnectionSee Example 3, page 349 • A rectangle is 4 times longer than it is wide. The perimeter of the rectangle is 30 cm. Find the dimensions of the rectangle.

  19. Summary • What did you learn today?

  20. SUMMARY • Solve for a variable. • Substitute. • Solve. • Substitute. • Three possible answers: • a point (x,y) • all real numbers • no solution

  21. ASSIGNMENT • 7.2, page 349, odds 1-39, & odds 47-59

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